Average Error: 33.5 → 10.1
Time: 9.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.17528679488360856 \cdot 10^{-69}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r30934 = b_2;
        double r30935 = -r30934;
        double r30936 = r30934 * r30934;
        double r30937 = a;
        double r30938 = c;
        double r30939 = r30937 * r30938;
        double r30940 = r30936 - r30939;
        double r30941 = sqrt(r30940);
        double r30942 = r30935 + r30941;
        double r30943 = r30942 / r30937;
        return r30943;
}
↓
double f(double a, double b_2, double c) {
        double r30944 = b_2;
        double r30945 = -4.032376794487168e+127;
        bool r30946 = r30944 <= r30945;
        double r30947 = 0.5;
        double r30948 = c;
        double r30949 = r30948 / r30944;
        double r30950 = r30947 * r30949;
        double r30951 = 2.0;
        double r30952 = a;
        double r30953 = r30944 / r30952;
        double r30954 = r30951 * r30953;
        double r30955 = r30950 - r30954;
        double r30956 = 1.1752867948836086e-69;
        bool r30957 = r30944 <= r30956;
        double r30958 = -r30944;
        double r30959 = r30944 * r30944;
        double r30960 = r30952 * r30948;
        double r30961 = r30959 - r30960;
        double r30962 = sqrt(r30961);
        double r30963 = sqrt(r30962);
        double r30964 = r30963 * r30963;
        double r30965 = r30958 + r30964;
        double r30966 = r30965 / r30952;
        double r30967 = -0.5;
        double r30968 = r30967 * r30949;
        double r30969 = r30957 ? r30966 : r30968;
        double r30970 = r30946 ? r30955 : r30969;
        return r30970;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -4.032376794487168e+127

    1. Initial program 53.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -4.032376794487168e+127 < b_2 < 1.1752867948836086e-69

    1. Initial program 12.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt12.7

      \[\leadsto \frac{\left(-b_2\right) + \sqrt{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Applied sqrt-prod12.9

      \[\leadsto \frac{\left(-b_2\right) + \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    if 1.1752867948836086e-69 < b_2

    1. Initial program 53.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 8.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))